When a shopkeeper gives 10% discount on the list price of a toy, his gain is 20%. If he had given a discount of 20%, his percentage of gain would have been

Options

A. `6(2)/(3)%`
B. `8(1)/(3)%`
C. `10%`
D. `15%`

Answer

To solve the problem step by step, let's break it down as follows:

### Step 1: Define Variables
Let the List Price (LP) of the toy be represented as \( LP \) and the Cost Price (CP) be represented as \( CP \).

### Step 2: Calculate Selling Price with 10% Discount
When the shopkeeper gives a 10% discount on the List Price:
- The Selling Price (SP) after a 10% discount is:
\[ SP = LP - 0.10 \times LP = 0.90 \times LP \]

### Step 3: Relate Selling Price to Cost Price
According to the problem, when the shopkeeper sells the toy at this Selling Price, he gains 20% profit on the Cost Price. Therefore:
- The Selling Price can also be expressed in terms of Cost Price:
\[ SP = CP + 0.20 \times CP = 1.20 \times CP \]

### Step 4: Set the Two Expressions for Selling Price Equal
From Steps 2 and 3, we have:
\[ 0.90 \times LP = 1.20 \times CP \]

### Step 5: Solve for the Ratio of List Price to Cost Price
Rearranging the equation gives:
\[ \frac{LP}{CP} = \frac{1.20}{0.90} = \frac{120}{90} = \frac{4}{3} \]
This means that the List Price is \( \frac{4}{3} \) times the Cost Price.

### Step 6: Calculate Selling Price with 20% Discount
Now, if the shopkeeper gives a 20% discount on the List Price:
- The new Selling Price (SP) will be:
\[ SP = LP - 0.20 \times LP = 0.80 \times LP \]

### Step 7: Substitute the List Price in Terms of Cost Price
Substituting \( LP \) in terms of \( CP \):
\[ SP = 0.80 \times \left(\frac{4}{3} \times CP\right) = \frac{3.2}{3} \times CP = \frac{32}{30} \times CP = \frac{16}{15} \times CP \]

### Step 8: Calculate the Gain
Now, we can find the gain:
- Gain = Selling Price - Cost Price
\[ Gain = SP - CP = \left(\frac{16}{15} \times CP\right) - CP = \left(\frac{16}{15} - \frac{15}{15}\right) \times CP = \frac{1}{15} \times CP \]

### Step 9: Calculate the Percentage Gain
To find the percentage gain:
\[ \text{Percentage Gain} = \left(\frac{Gain}{CP}\right) \times 100 = \left(\frac{\frac{1}{15} \times CP}{CP}\right) \times 100 = \frac{1}{15} \times 100 = \frac{100}{15} \approx 6.67\% \]

### Final Answer
Thus, if the shopkeeper gives a 20% discount, his percentage of gain would be approximately \( 6.67\% \) or \( 6 \frac{2}{3}\% \).

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